Today in Astronomy 102: general relativity and the
prediction of the existence of black holes
 Experimental tests:
Einstein’s theories
are proven to be
valid.
 Schwarzschild solves
Einstein’s equations,
applied to stars, and
finds black holes in
the results.
A
G
B
Image: gravitational lens in galaxy cluster AC114, by the Richard Ellis
and Ian Smail with the Hubble Space Telescope. Note that the “quasars”
A and B are mirror images of each other.
2 October 2001
Astronomy 102, Fall 2001
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Experimental tests of relativity
Einstein’s theories of relativity represent a rebuilding of
physics from the ground up.
 New postulates and assumptions relate the most basic
concepts, such as the relativity of space and time and the
“absoluteness” of the speed of light.
 The new theory is logically consistent and mathematically
very elegant.
 The new theory “contains” classical physics as an
approximation valid for speeds much smaller than the
speed of light.
 Still: it would be worthless if it didn’t agree with reality
(i.e. experiments) better than classical physics.
2 October 2001
Astronomy 102, Fall 2001
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Experimental tests of scientific theories
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Speed (cm/sec)
No scientific theory is valid unless:
 it is mathematically and
logically consistent,
 and its predictions can be
measured in experiments,
 and the theoretical predictions
are in precise agreement with
experimental measurements.
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Example: prediction by the “old” and
Time since start (sec)
Galilean relativity
“new” relativity theories for the speed
Special theory of relativity
of a body accelerated with constant
Measurements
.
power. Both theories are valid for low speeds but only
special
relativity is valid over the whole range of measurements.
2 October 2001
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Experimental tests of scientific theories
(continued)
2 October 2001
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Speed (cm/sec)
What constitutes a valid scientific
experiment?
 Measurements are made with
accuracy (the size of potential
measurement errors) sufficient
to test the predictions of
prevailing theories,
 and the accuracy can be
estimated reliably,
 and the measurements are
reproducible: the same results
are obtained whenever, and by
whomever, the experiments are
repeated.
Astronomy 102, Fall 2001
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Time since start (sec)
Galilean relativity
Special theory of relativity
Measurements
.
4
Experimental tests of relativity (continued)
Some experiments testing special relativity:
 Michelson or Michelson-Morley type: speed of light same
in all directions, to extremely high accuracy.
 High-energy accelerators used in elementary particle
physics:
• radioactive particles are seen to live much longer when
moving near light speeds than when at rest (direct
observation of time dilation).
• though accelerated particles get extremely close to the
speed of light, none ever exceed it.
 Nuclear reactors/bombs: mass-energy equivalence
(E=mc2).
2 October 2001
Astronomy 102, Fall 2001
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Every day, many millions of special-relativity
experiments
Fermi National
Laboratory, near
Chicago, currently
the world’s highestenergy elementaryparticle accelerator.
2 October 2001
Astronomy 102, Fall 2001
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Experimental tests of relativity (continued)
Some experiments testing general relativity:
 Gravitational lensing: light following gravity-induced
curves in spacetime.
• Starlight deflected by the sun
• Light from distant quasars deflected by galaxies
 Precession of the “perihelion” of Mercury’s orbit: matter
following gravity-induced curves in spacetime.
 Gravitational redshifts on Earth and in the spectrum of
the sun: direct observation of gravitational time dilation.
 Discovery of gravitational radiation (Nobel Prize in
physics, 1993, Taylor and Hulse).
 Discovery of black holes (as we’ll see).
2 October 2001
Astronomy 102, Fall 2001
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Gravitationally-lensed quasars
Images and
diagram by
Alan Stockton,
U. Hawaii.
“Double quasar:”
mirror image of A
subtracted from B
leaves a faint galaxy.
2 October 2001
Light from distant quasar Q follows
warped space around galaxy G; we
(at O) see images of Q in two
different places on the sky. (Vertical
scale greatly exaggerated.)
Astronomy 102, Fall 2001
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Experimental tests of relativity (continued)
Results of experiments:
 All reproducible experiments to date have confirmed the
predictions of Einstein’s relativity theories.
 Few scientific theories are so well-supported by
experiment, in fact.
 We keep using the theory to predict new effects. Those
effects involving conditions within those for which the
theory has been tested are very likely to be real.
Experimental tests of these newly-predicted effects are in
many cases even sterner tests of the theories.
Black holes were among the first of these “new effects”
predicted by the general theory of relativity, though this was
not recognized at the time.
2 October 2001
Astronomy 102, Fall 2001
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Mid-lecture break.
Exam #1 will take place here in Hubbell Auditorium on
Thursday, 4 October 2001, 2-3:15 PM.
 Be sure you check out the Practice Exam and its solutions,
both of which are available in the AST 102 Web Site.
 The TAs will be running a Review Session, tomorrow at 8
PM (that is, after sunset), here in Hubbell Auditorium.
Bring a calculator, a writing instrument, and some experience
solving problems such as have been on the homework and
Practice Exam. Everything else you need will be provided,
including formulas and a copy of the How Big Is That sheet.
2 October 2001
Astronomy 102, Fall 2001
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Mid-lecture break.
Homework hints
 Problem 1
The terms are left loosely defined:
“slightly, but noticeably” means different by 0.1% to a few
% or so;
“substantially” means different by a large percentage or
by a factor greater than 1;
“equal, to high accuracy” means exactly equal or so close
that it takes an extraordinary experimental effort to tell
the difference;
“infinite” means infinite.
2 October 2001
Astronomy 102, Fall 2001
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Karl Schwarzschild’s Work
Schwarzschild was interested in the
physics of stars, so he solved Einstein’s
field equation for the region outside a
massive spherical object.
 In the process he discovered some
techniques, and ways of visualizing
curved spacetime, that benefited
others who were doing research in
general relativity.
• Embedding diagrams
 His solution of the Einstein field
equation revealed the existence of
black holes.
Karl Schwarzschild
2 October 2001
Astronomy 102, Fall 2001
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Einstein’s field equation
The field equation is the ultimate mathematical expression of
Einstein’s general theory of relativity.
Astronomy 102 version:
“Spacetime, with its curvature, tells masses how to move;
masses tell spacetime how to curve.”
Physics 413 - Astronomy 554 version:
 2 l
 x h  2  a x s  2  a
 2 g ml
 2 g mk 
 2 g lk
x h  2  a x s  2  a 
1   gl
 + g hs 
+
- a

a
l
a
m k
k
a
m l
2  x k x m x k x l x l x m x l x l 













x xl
x x
x xl
x x 


  2 gl
h
  h 2a  s 2a
2  a  s  2  a 
 2 g lm 
 2 g lm


1
1
x
x
x
x
l

 + g hs 

- g mk
- m
- a
2 m
a
l
a
m
m
a
l

x l x m 
  x x l  x m x
2
 x x l  x m x 
 2  x x m x x l


= -8pG
p
n
2 October 2001
nm
dx nk 3
d (x - xn )
dt
Astronomy 102, Fall 2001
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What you get when you solve the field equation
In case you’re interested (i.e. not on the exam):
 The solution to the field equation is a function called the
metric tensor. This function tells how much distance or
time is involved for unit displacements in the spacetime
coordinates.
 Accordingly, the metric tensor is related to the absolute
interval. Each different solution to the field equation
corresponds to a different absolute interval. The absolute
interval corresponding to Schwarzschild’s metric turns
out to be given (in spherical coordinates) by
2
D
2GM
r
2
Ds2 =
+ r 2 Dq 2 + r 2 sin 2 qDf 2 - c 2 1 D
t
2
2GM
rc
1rc 2
F
H
2 October 2001
Astronomy 102, Fall 2001
I
K
14
Embedding diagrams, and why they’re useful
Embedding diagrams provide an analogy by which one can
envision how four-dimensional spacetime can be warped, by
introduction of additional, imaginary dimensions. These
additional dimensions are called hyperspace.
First, consider circles in flat two-dimensional space:
C = 2pr = pd
C
r
2 October 2001
Astronomy 102, Fall 2001
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Embedding
diagrams, and
why they’re
useful
(continued)
C
Circle drawn
in rubber sheet
Figure from
Thorne, Black holes
and time warps
Stretch!
2 October 2001
r
An embedding diagram.
Circles in curved space still
look like 2-D figures, but
behave as if their centers are
much further away than
C/2p. We can picture this as
a stretching of space in the
direction perpendicular to
the circle.
Astronomy 102, Fall 2001
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Embedding diagrams, and why they’re useful
(continued)
C = pd if spacetime is flat
(i.e. if gravity is weak)
Figure from
Thorne, Black holes
and time warps
2 October 2001
C < pd if spacetime
is curved (i.e. if
gravity is strong)
Astronomy 102, Fall 2001
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Schwarzschild’s view of the
equatorial plane of a star
Figure from
Thorne, Black holes
and time warps
(equator)
Center of star
2 October 2001
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Schwarzschild’s solution to the Einstein field
equations
Results: the curvature of spacetime outside a massive star.
 For a given, fixed star mass M, he examined how space
and time are curved if the star is made smaller and
smaller in size.
 Singularity: if the star is made smaller than a certain
critical size, the gravitational redshift of light (time
dilation, remember) predicted by his solution was infinite!
(See next slide.)
2 October 2001
Astronomy 102, Fall 2001
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Figure from Thorne, Black
holes and time warps
2 October 2001
Think of this as light losing energy, as it
crawls out of a deeper and deeper hole.
Astronomy 102, Fall 2001
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Implications of “Schwarzschild’s singularity”
 If a star is made too small in circumference for a given
mass, nothing can escape from it, not even light.
• This would be a black hole, and the critical size is the
size of the black hole’s horizon.
 This is similar to an 18th century idea: “dark stars”
(Michell, Laplace...); that if light were subject to
gravitational force, there could be stars from which light
could not escape.
• The critical size of Schwarzschild’s singularity turns
out to be the same as that for the 18th century dark
star.
2 October 2001
Astronomy 102, Fall 2001
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